Dear Fred, Such a T is called a symmetric monoidal functor. Example: let _A_ be Set with the cartesian monoidal structure. Let M be a monoid and let T be the functor Set->Set sending X to MxX (which I'll write as MX). This functor T is monoidal via the map MXMY->MXY sending (m,x,n,y) to (mn,x,y). It is symmetric monoidal iff M is commutative. Steve Lack. On 6/05/10 4:01 PM, "Fred E.J. Linton" <fejlinton@usa.net> wrote:
Suppose _A_ is a symmetric monoidal category in the sense of the Eilenberg-Kelley La Jolla paper, and T: _A_ --> _A_ a monoidal functor.
What, if anything, is known, where τ: X ⊗ Y --> Y ⊗ X is the symmetry structure on the (symmetric) tensor product ⊗, as to whether
[T_X,Y: TX ⊗ TY --> T(X ⊗ Y)] and [T(τ_X,Y): T(X ⊗ Y) --> T(Y ⊗ X)]
have the same composition as have
[τ_TX,TY: TX ⊗ TY --> TY ⊗ TX] and [T_Y,X: TY ⊗ TX --> T(Y ⊗ X)] ?
TIA for any relevant information and/or references thereto.
Cheers, -- Fred
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